# finding process corresponding to laplace transform

I have a positive stochastic process $$X(t)$$ with Laplace transforms $$\mathbb{E}\left[\mathrm{e}^{-uX(t)}\right]=\left(\frac{a+u\mathrm{e}^{-\kappa t}}{a+u}\right)^{b}$$ One can clearly see that the stationary distribution of the process as $$t\rightarrow\infty$$ is Gamma-distributed. However, I'm interested in the distribution at a finite time $$t$$. Is there a known distribution (or transformation of a known distribution) that has this Laplace transform?